UPSC NDA I 2024 Mathematics Paper

Concept:Analyze monotonicity of $f(x) = \frac{x}{\ln x}$ using derivative sign and compare logarithmic expressions.Explanation:Given $f(x) = \frac{x}{\ln x}$ for $x>1$.Compute $f'(x) = \frac{(\ln x) - 1}{(\ln x)^2}$ using quotient rule.For statement 1: In $(e, \infty)$, $\ln x > 1$, so $\ln x - 1 > 0$. Thus $f'(x) > 0$, so $f(x)$ is increasing. Correct.For statement 2: In $(1, e)$, $0 < \ln x < 1$, so $\ln x - 1 < 0$. Thus $f'(x) < 0$, so $f(x)$ is decreasing. Correct.For statement 3: Compare $9\ln 7$ and $7\ln 9$.Rewrite as $\ln(7^9)$ and $\ln(9^7)$. Since $\ln$ is increasing, compare $7^9 = 40353607$ and $9^7 = 4782969$.Clearly $7^9 > 9^7$, so $\ln(7^9) > \ln(9^7)$, i.e. $9\ln 7 > 7\ln 9$. Correct.Answer:1, 2 and 3 all are correct. Option D.